normal distribution

The symmetric bell-shaped distribution; about 68% of values lie within 1 SD of the mean and about 95% within 2 SD.

The normal distribution is the classic bell curve: symmetric, with a single peak in the middle and thin tails that fade out on both sides. Two numbers describe it completely: the mean $\mu$ sets where the peak sits, and the standard deviation $\sigma$ sets how wide the bell is. The mean, median and mode all coincide at the centre. The curve's formula is

$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

but in practice you almost never need it. What you need is its consequence, the 68–95–99.7 rule: about 68% of values fall within $\mu \pm \sigma$ , about 95% within $\mu \pm 2\sigma$ and about 99.7% within $\mu \pm 3\sigma$ . (The exact 95% boundary is $1.96\sigma$ , the number behind most confidence intervals.) Convert any value to a z-score and one standard table works for every normal variable.

Where does it appear? In quantities shaped by many small, independent nudges, such as the fill weight of a food bag or measurement noise. For PMs the more important place is one step removed: thanks to the central limit theorem, sample means and conversion rates from reasonably large samples are close to normal even when the raw data are not. That is why z-tests, t-tests and confidence intervals lean on the bell curve.

What is usually not normal: revenue per customer, order value, session length, anything with a floor at zero and a long right tail. Don't assume the shape; check it with a histogram.

Example

CatChow's factory fills standard 2 kg bags. Fill weights are close to normal with mean $\mu = 2{,}000$ g and SD $\sigma = 10$ g.

By the 68–95–99.7 rule:

Range	Weights	Share of bags
$\mu \pm 1\sigma$	1,990–2,010 g	about 68%
$\mu \pm 2\sigma$	1,980–2,020 g	about 95%
$\mu \pm 3\sigma$	1,970–2,030 g	about 99.7%

The label promises at least 1,980 g. How many bags in a batch of 5,000 fall short? 1,980 g is exactly $2\sigma$ below the mean. About 95% of bags are within $\pm 2\sigma$ , so the remaining 5% split equally between the two tails: roughly 2.5% are too light. The exact table value for $z = -2$ is 2.28%, so $0.0228 \times 5{,}000 \approx 114$ bags.

Now suppose the filling machine gets sloppier and $\sigma$ grows to 15 g while the mean stays at 2,000 g. The same 1,980 g is now $z = (1{,}980 - 2{,}000) / 15 \approx -1.33$ , and about 9.1% of bags fall below it: roughly 456 of 5,000. The average didn't move at all, yet underweight bags quadrupled. With normal data, the spread matters as much as the mean.

Common mistakes

Assuming revenue or order value is normal. Money metrics are usually right-skewed; the bell curve will understate how often big orders happen.
Thinking every test needs normal raw data. For means of large samples the central limit theorem does that job; normality of the raw data matters mainly for small samples.
Using the 68–95–99.7 rule to flag outliers in skewed data. The rule only holds for bell-shaped data.
Reading "normal" as "healthy" or "typical". It is the name of a shape, not a verdict on your metric.
Trusting a formal normality test on a huge sample. With 100,000 rows it rejects normality for harmless wiggles; look at the histogram and ask whether the deviation changes your decision.

Learn it in the course

Samples, populations and the normal distribution · Why we study samples to learn about all users, and why the bell curve shows up everywhere.
Confidence intervals · Reporting a range of plausible values instead of a single number.